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A series expansion for standard normal probabilities

Published online by Cambridge University Press:  17 February 2025

Geoffrey W. Brown  and
Adam C. Brown
Geoffrey W. Brown
Affiliation:
Toronto Metropolitan University, 350 Victoria Street, Toronto, ON M5B 2K3 e-mail: [email protected]
Adam C. Brown
Affiliation:
University of Toronto Schools, 371 Bloor St W, Toronto, ON M5S 2R7 e-mail: [email protected]
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Extract

The goal of this paper is to derive a series expansion for standard normal probabilities F(b), where b ≥ 0 and the upper half of the cumulative distribution function (or CDF) of the standard normal is given by(1) See [1, Chapter 4] for information and results about the normal distribution. It is known that this integral is not elementary, and the usual approach is to use the Taylor series expansion for ez in order to evaluate the integral. We know that and therefore that (2).

Type
Articles
Information
The Mathematical Gazette , Volume 109 , Issue 574 , March 2025 , pp. 17 - 26
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Copyright
© The Authors, 2025 Published by Cambridge University Press on behalf of The Mathematical Association

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References

Mathematical statistics with applications, (7th edn), Dennis D. Wackerly, William Mendenhall III, Richard L. Scheaffer, Brooks/Cole Cengage Learning (2008).Google Scholar
Normal distribution, Wikipedia, accessed July 2024 at https://en.wikipedia.org/wiki/Normal_distribution Google Scholar
CRC Standard Mathematical Tables and Formulae (31st edn), Daniel Zwillinger, Chapman & Hall, (2003).Google Scholar